The Hamiltonian Limit and Universality

Malte Henkel

The scaling amplitudes of the spin-spin and energy-energy correlation lengths for the Hamiltonian limit of the 3D Ising model are computed on a square lattice. Both periodic and antiperiodic boundary conditions are considered. The ratios of the scaling amplitudes are found to be universal.

P.J. Morrison, C. S. Jones

Jacob Biamonte, Peter J. Love

It has been established that local lattice spin Hamiltonians can be used for universal adiabatic quantum computation. However, the two-local model Hamiltonians used in these proofs are general and hence do not limit the types of interactions required between spins. To address this concern, the present paper provides two simple model Hamiltonians that are of practical interest to experimentalists working toward the realization of a universal adiabatic quantum computer. The model Hamiltonians presented are the simplest known quantum-Merlin-Arthur-complete (QMA-complete) two-local Hamiltonians. The two-local Ising model with one-local transverse field which has been realized using an array of technologies, is perhaps the simplest quantum spin model but is unlikely to be universal for adiabatic quantum computation. We demonstrate that this model can be rendered universal and QMA-complete by adding a tunable two-local transverse ${\ensuremath{\sigma}}^{x}{\ensuremath{\sigma}}^{x}$ coupling. We also show the universality and QMA-completeness of spin models with only one-local ${\ensuremath{\sigma}}^{z}$ and ${\ensuremath{\sigma}}^{x}$ fields and two-local ${\ensuremath{\sigma}}^{z}{\ensuremath{\sigma}}^{x}$ interactions.

A. Horzela

Stanisław D. Głazek, Kenneth Geddes Wilson

The universality of renormalization-group limit-cycle behavior is illustrated with a simple discrete Hamiltonian model. A nonperturbative renormalization-group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound-state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero---also a consequence of the limit cycle. Wegner's eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegner's eigenvalues are independent of location on the limit cycle, in contrast with Wegner's operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonian's bound-state eigenvalues converge much more quickly than otherwise to an exact geometric series.